Chapter 5
The Binomial Probability Distribution and Related Topics
(S)
1. Identify each of the random variables as continuous or discrete.
(a) The number of cows in a pasture
(b) The number of electrons in a molecule
(c) The voltage on a power line
(d) The volume of milk given by a cow per milking
(e) The distance from Cape Canaveral to a point chosen at random in the Sea of
Tranquillity on the moon
(f) The number of words in a book chosen at random
(S)
2. Identify each of the random variables as continuous or discrete.
(a) The time in hours that you sleep on a random weekday night
(b) The home team score in a basketball game
(c) The number of ducks on a pond
(d) A score on the reading comprehension portion of the SAT exam
(e) The volume of water in Lake Powell
(f) The number of fish in Lake Lulu
(S)
3. At Cape College the business students run an investment club. Each fall they create
investment portfolios in multiples of $1,000 each. Records from the past several
years show the following probabilities of profits (rounded to the nearest $50). In
the table below, x = profit per $1,000 and P(x) is the probability of earning that
profit.
x
0
50
100
150
200
P(x)
0.15
0.35
0.25
0.20
0.05
(a) Find the expected value of the profit in a $1,000 portfolio.
(b) Find the standard deviation of the profit.
(c) What is the probability of a profit of $150 or more in a $1,000 portfolio?
(S)
4. A franchise chain of small grocery stores has kept records of the number of bad
checks passed in its stores. They used the data to get a probability distribution for
the number of bad checks passed in a store each week. In the table below x =
number of bad checks and P(x) is the probability that x bad checks will be passed in
a week.
x
0
1
2
3
4
P(x)
0.3
0.3
0.2
0.1
0.1
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127
128
Test Item File Understandable Statistics, 7th Edition
(a)
(b)
(c)
(d)
(S)
Calculate the expected number of bad checks the chain will get in one week.
Calculate the standard deviation for the number of bad checks.
What is the probability that two or more bad checks will be passed in a week?
What is the probability that no bad checks will be passed in a week?
5. George is a computer salesman who usually visits 6 customers each day. Over the
years, George has recorded the number of sales per day. He used this data to
estimate the probability of 0 sales, 1 sale, 2 sales and so on up to 6 sales per day. In
the table below, x = number of sales per day and P(x) is the probability of making x
sales.
x
0
1
2
3
4
5
6
P(x)
0.047
0.187
0.311
0.276
0.138
0.037
0.004
(a) The number of sales can be thought of as a random variable. Is it discrete or
continuous? Explain.
(b) Compute the expected value for the number of sales on a typical day.
(c) Compute the standard deviation for the number of sales.
(S)
6. Carol is a dental assistant. She usually has five clients per day. Over the years, she
has recorded the number of cancellations and used the data to estimate the
probability of 0 through 5 cancellations per day. In the table below x = number of
cancellations per day and P(x) = the probability of x cancellations.
x
0
1
2
3
4
5
P(x)
0.328
0.410
0.205
0.051
0.005
0.001
(a) The number of cancellations is a random variable. Is it discrete or continuous?
Explain.
(b) Compute the expected number of cancellations per day.
(c) Compute the standard deviation for the number of cancellations.
(S)
7. A local cab company is interested in the number of pieces of luggage a cab carries
on a taxi run. A random sample of 260 taxi runs gave the following information.
x = number of pieces of luggage and f is the frequency with which taxi runs carried
x pieces of luggage.
x
0
1
2
3
4
5
6
7
8
9
10
f
42
51
63
38
19
16
12
10
6
2
1
(a) Find the probability distribution for x.
(b) Make a histogram of the probability distribution.
(c) Estimate the probability that a taxi run will have from 0 to 4 pieces of luggage
(including 0 and 4).
(d) Compute the expected value of x.
(e) Compute the standard deviation for x.
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Chapter 5 The Binomial Probability Distribution and Related Topics
(S)
129
8. The Army gives a battery of exams to all new recruits. One exam measures a
person’s ability to work with technical machinery. This exam was given to a random
sample of 360 new recruits. In the table below x = score on the test and f is the
frequency of new recruits with score x.
x
1
2
3
4
5
6
7
8
9
10
f
28
42
79
83
51
36
18
12
7
4
(a) Find the probability distribution for these scores.
(b) Draw a histogram of the probability distribution.
(c) The ground-to-air missile battalion needs people with a score of 7 or higher on
the exam. What is the probability that a new recruit will meet this criterion?
(d) The kitchen battalion can use people with scores of three or less. What is the
probability that a new recruit is not overqualified for this work?
(e) Compute the expected value of these scores.
(f) Compute the standard deviation of these scores.
(S)
9. Reggie Richman has a poor driving record and must take out a special insurance
policy. He wants to insure his $58,000 sports car. Based on previous driving
records, AnyState has estimated the probability of various levels of loss per year as
indicated in the table below. x = dollar amount of loss. The company will pay no
benefits for any other partial losses.
% of loss
100
50
25
10
0
x
58,000
29,000
14,500
5,800
0
P(x)
0.05
0.12
0.28
0.42
0.13
(a) Calculate the expected loss on Reggie’s car.
(b) How much should AnyState charge Reggie for insurance if it wants to make a
profit of $1,000 on his policy.
(S) 10. Mr. Dithers wants to insure his yacht for $280,000. The Big Rock insurance
company estimates that, over a period of one year, a total loss may occur with
probability 0.005, a 50% loss with probability 0.01, a 25% loss with probability
0.05, a 10% loss with probability 0.08. Big Rock will pay for no other partial losses.
(a) What is the expected loss on this yacht?
(b) How much should the company charge Mr. Dithers for insurance if it wants to
make a profit of $1000 per year on his policy?
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130
Test Item File Understandable Statistics, 7th Edition
(S) 11. The student senate is sponsoring a car raffle to raise money to buy playground
equipment for disadvantaged children. The senate received a donated car worth
$7,000 and sold 3750 raffle tickets at $5 each.
(a) If you buy 10 tickets, what is the probability that you will win the car? What is
the probability that you will not win the car?
(b) Compute your expected earnings. (Your expected earnings is the product of the
value of the car and the probability that you will win it.)
(c) How much did you contribute to the playground fund? (The amount of the
contribution is the difference between what you paid for the tickets and your
expected earnings.)
(S) 12. The Old West Historical Museum is trying to raise money for building restoration.
A travel agent donated an all-expense paid trip to Hawaii for two, valued at
$3587.00. Friends of the Museum sold 1950 tickets at $5.00 per ticket.
(a) If you buy 8 tickets, what is the probability that you will win the trip? What is
the probability that you will not win the trip?
(b) What are your expected earnings from this raffle?
(c) How much do you contribute to the Museum through your ticket purchase?
(The amount contributed is the difference between the cost of the tickets and
your expected earnings.)
(S) 13. To estimate the number of books required for a course, the student council at Lees
Field College took a random sample of 100 courses and obtained the information
below. x = number of books and f = number of courses requiring x books.
x
0
1
2
3
4
5
6
f
5
48
22
11
9
3
2
(a) Make a table showing the probability distribution for the number of required
books.
(b) Make a histogram of the probability distribution.
(c) Find the expected number of books per course and the standard deviation.
(d) If the average book costs $42 and you take 5 courses, how much can you expect
to spend for books?
(M) 14. Identify which one of the following random variables is continuous.
A. The number of fish caught by a fishing boat
B. The number of coins contained in a slot machine
C. The number of traffic accidents in a city
D. The number of gallons of water in a reservoir
E. The number of tables sold at a furniture store
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Chapter 5 The Binomial Probability Distribution and Related Topics
131
(M) 15. Identify which one of the following random variables is discrete.
A. The number of inches of rainfall in a county
B. The number of beverages sold at a lemonade stand
C. The number of gallons of concrete used at a construction site
D. The time required for a runner to finish a marathon
E. The temperature of a pot roast cooking in an oven
(S) 16. Identify each random variable as continuous or discrete.
(a) Speed of an automobile
(b) The number of doughnuts left in the pantry
(c) The air temperature of a public park
(d) The weight of a professional wrestler
(e) The number of restaurant patrons
(M) 17. Choose the answer below that identifies a value for y that results in a valid
probability distribution.
x
0
1
2
P(x)
0.35
0.5
y
A.
B.
C.
D.
E.
y = –1.85
y = 0.6
y = 1.85
y = 0.06
y = 0.15
(M) 18. A police commissioner is interested in the number of phone calls into the police
dispatcher that result in criminal charges actually being filed. For a random sample
of 190 nights (7 PM to 7 AM), the following information was obtained where x =
number of calls resulting in criminal charges and f = frequency with which this
many calls occurred (i.e., number of nights).
x
25
26
27
28
29
30
31
32
33
34
f
11
16
17
22
24
17
23
26
21
13
Assuming these 190 nights represent the population of all nights in this police
precinct, what do you estimate the probability is that on a randomly selected night,
there will be from 27 to 31 (including 27 and 31) calls to the dispatcher that result in
criminal charges being filed?
A. 0.21
B. 0.33
C. 0.46
D. 0.54
E. 0.67
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132
Test Item File Understandable Statistics, 7th Edition
(M) 19. A police commissioner is interested in the number of phone calls into the police
dispatcher that result in criminal charges actually being filed. For a random sample
of 190 nights (7 PM to 7 AM), the following information was obtained where x =
number of calls resulting in criminal charges and f = frequency with which this
many calls occurred (i.e., number of nights).
x
25
26
27
28
29
30
31
32
33
34
f
11
16
17
22
24
17
23
26
21
13
Assuming these 190 nights represent the population of all nights in this police
precinct, what do you estimate the probability is that on a randomly selected night,
there will be from 25 to 29 (including 25 and 29) calls to the dispatcher that result in
criminal charges being filed?
A. 0.18
B. 0.28
C. 0.29
D. 0.47
E. 0.71
(M) 20. A police commissioner is interested in the number of phone calls into the police
dispatcher that result in criminal charges actually being filed. For a random sample
of 190 nights (7 PM to 7 AM), the following information was obtained where
x = number of calls resulting in criminal charges and f = frequency with which this
many calls occurred (i.e., number of nights).
x
25
26
27
28
29
30
31
32
33
34
f
11
16
17
22
24
17
23
26
21
13
Assuming these 190 nights represent the population of all nights in this police
precinct, use relative frequencies to find P(x) when x = 25, 26, 27, 28, 29, 30, 31,
32, 33, and 34 rounded to two decimal places. Use these probabilities to find the
expected number of calls to the dispatcher that result in criminal charges being filed.
A. 20.16
B. 23.12
C. 27.09
D. 29.75
E. 32.13
(S) 21. For a particular lake in the U.S., the following table shows the percentage of
fishermen who caught x number of fish during a 6-hour period while fishing from
shore.
x
0
1
2
3 or more
%
52
27
14
7
(a) Find the probability that a fisherman selected at random fishing from shore
catches one or more fish during a 6-hour period.
(b) Find the probability that a fisherman selected at random fishing from shore
catches two or more fish during a 6-hour period.
(c) Compute µ, the expected value of the number of fish caught per fisherman in a
6-hour period (round 3 or more to 3). Round to two decimal places.
(d) Compute ! , the standard deviation of the number of fish caught per fisherman
in a 6-hour period (round 3 or more to 3). Round to two decimal places.
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Chapter 5 The Binomial Probability Distribution and Related Topics
133
(S) 22. Eric teaches ceramics in his studio. He estimates that one out of every five people
who call for information about a class will sign up for the class. Last week he
received nine calls. Find the probability that four or fewer of the people who called
will sign up for a class.
(S) 23. Dorothy leads wilderness expeditions. She has found that 15% of those who attend a
promotional meeting will sign up for an expedition at the meeting. If 20 people
attend a meeting:
(a) What is the probability that none of them will sign up?
(b) What is the probability that five or more of them will sign up?
(S) 24. In Summit County 65% of the voter population are Republicans. What is the
probability that a random sample of ten Summit County voters will contain:
(a) Exactly eight Republicans?
(b) Exactly two Republicans?
(c) No Republicans?
(d) All Republicans?
(S) 25. The college health center did a campus-wide survey of students and found that 15%
of the students smoke cigarettes. A group of nine students randomly come together
and sit at the same table on the plaza in front of the library. Find the probability that:
(a) No student at the table smokes.
(b) At least one student at the table smokes.
(c) More than two students smoke.
(d) From one to five smoke (including one and five).
(S) 26. The mayor said that local fire stations are so well equipped and staffed that they are
able to get to a fire in five minutes or less about 80% of the time. If this is so, find
the probability that out of the next ten fire calls:
(a) The fire department arrives in five minutes or less for every call.
(b) It takes five minutes or less for five or more of the calls.
(c) It takes the fire department more than five minutes to arrive for all ten calls.
(d) For at least three calls it takes more than five minutes for the fire department to
arrive.
(S) 27. A TV sports commentator claims that 45% of all football injuries are knee injuries.
Assuming that this claim is true, what is the probability that in a game with 5
reported injuries:
(a) All are knee injuries?
(b) None are knee injuries?
(c) At least three are knee injuries?
(d) No more than 2 are knee injuries?
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134
Test Item File Understandable Statistics, 7th Edition
(S) 28. A safety engineer claims that 18% of all automobile accidents are due to
mechanical failure. Assume that this is correct and use the formula for binomial
probabilities to find the probability that exactly three out of eight automobile
accidents is due to mechanical failure.
(S) 29. Martha estimates the probability that she will receive at least one marketing
telephone call at home during the hours of 5pm to 7pm on a weekday night to be
2/3. Use the formulas for computing binomial probabilities to answer the following
questions:
(a) What is the probability that she will receive at least one call on all five of the
next five weekday nights?
(b) What is the probability that she will not receive a call on any of the next five
weekday nights?
(c) What is the probability that she will receive a call on at least four of the next
five weekday nights?
(S) 30. It is estimated that four out of five patients who have an artery bypass heart
operation survive eight years without needing another bypass operation. Of seven
patients who recently had such an operation, what is the probability that:
(a) All will survive eight years without needing another bypass operation?
(b) At least four will survive eight years without needing another bypass operation?
(c) Exactly four will survive eight years without needing another bypass operation?
(S) 31. The probability of an adverse reaction to a pneumonia shot is 0.15. Pneumonia
shots are given to a group of 12 people.
(a) What is the probability that none of them will have an adverse reaction to the
pneumonia shot?
(b) What is the probability that more than half of them will have an adverse
reaction to the pneumonia shot?
(c) What is the probability that 1 or fewer of the 12 people have an adverse reaction
to the pneumonia shot?
(d) What is the probability that two or more of them will have an adverse reaction
to the pneumonia shot?
(S) 32. The probability that a restaurant patron will request seating in the outdoor patio is
0.45. A random sample of 7 people call to make reservations.
(a) Find the probability that fewer than three of them request outdoor seating.
(b) Find the probability that three or more of them request outdoor seating.
(c) Find the probability that none of them request outdoor seating.
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Chapter 5 The Binomial Probability Distribution and Related Topics
135
(S) 33. It is claimed that 70% of the cars on the Valley Highway are going faster than 65
miles per hour. A random sample of 12 cars was observed under normal driving
conditions with no police car in sight.
(a) What is the probability that all of them were going faster than 65 miles per
hour?
(b) What is the probability that fewer than half of them were going over 65 miles
per hour?
(S) 34. A fair quarter is flipped three times.
(a) Find the probability of getting exactly three heads.
(b) Find the probability of getting two or more heads.
(c) Find the probability of getting fewer than two heads.
(d) Find the probability of getting exactly two heads.
(S) 35. A fair quarter is flipped 11 times.
(a) Find the probability of getting exactly 6 heads.
(b) Find the probability of getting more than 6 heads.
(c) Find the probability of getting fewer than 6 heads.
(d) Find the probability of getting four to nine heads (including four and nine).
(S) 36. Richard has just been given a ten-question multiple choice test in his history class.
Each question has five answers only one of which is correct. Since Richard has not
attended class recently, he does not know any of the answers. Assume that Richard
guesses randomly on all ten questions.
(a) Find the probability that he will answer all ten questions correctly.
(b) Find the probability that he will answer five or more questions correctly.
(c) Find the probability that he will answer none of the questions correctly.
(d) Find the probability that he will answer at least three questions correctly.
(S) 37. A tourist bureau for a western state conducted a study which showed that 65% of
the people who seek information about the state actually come for a visit. The office
receives 15 requests for information on the state.
(a) Find the probability that all 15 of the people visit the state.
(b) Find the probability that at least 9 of the people visit the state.
(c) Find the probability that no more than 8 of the people visit the state.
(d) Find the probability that from 4 to 10 people visit the state (including 4 and 10).
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136
Test Item File Understandable Statistics, 7th Edition
(S) 38. A certain type of penicillin will cause a skin rash in 10% of the patients receiving it.
(a) If the penicillin is given to a random sample of 15 patients what is the
probability that no more than 2 will have a skin rash?
(b) If it is given to a random sample of 9 patients, what is the probability that at
least one will have a skin rash?
(c) If it is given to a random sample of 12 patients what is the probability that more
than two will have a skin rash?
(S) 39. A recent report stated that 35% of teenagers of driving age owned a car. Consider a
random sample of eight teenagers of driving age.
(a) Find the probability that five or more of them do not own a car.
(b) Find the probability that three or fewer of them do own a car.
(S) 40. A recent medical survey reported that 45% of the respondents to a poll on patient
care felt that doctors usually explain things well to their patients. Assuming that the
poll reflects the feelings of all patients toward their doctors, find the probability that
for 12 patients selected at random:
(a) Four of more agree with the statement.
(b) No more than 8 do not agree with the statement.
(S) 41. Sam is a computer salesman who has a history of making successful calls one-fourth
of the time.
(a) What is the probability that he will be successful on at least three of the next
five calls?
(b) What is the probability that he will be successful on none of the next five calls?
(M) 42. In rolling two fair dice simultaneously, the probability of rolling a 7 or 11 is 29 .
What is the approximate probability of rolling a 7 or 11 two or more times in ten
rolls of two dice?
A. 0.08
B. 0.23
C. 0.30
D. 0.69
E. 0.77
(M) 43. In rolling two fair dice simultaneously, the probability of rolling a 7 or 11 is 29 .
What is the approximate probability of rolling a 7 or 11 at least once in 5 rolls of
two dice?
A. 0.28
B. 0.41
C. 0.63
D. 0.69
E. 0.72
(M) 44. In rolling two fair dice simultaneously, the probability of rolling “snake eyes”, that
1
is, a 1 on each die for a sum of 2, is 36
. What is the probability of rolling either 1 or
2 snake eyes in 10 rolls of two dice?
A. 0.22
B. 0.24
C. 0.25
D. 0.31
E. 0.32
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Chapter 5 The Binomial Probability Distribution and Related Topics
137
(M) 45. Assume that 67% of the cars on a particular freeway are traveling faster than 70
miles per hour. A random sample of 15 cars was observed under normal driving
conditions with no police car in sight. What is the probability that 8 or more of them
were going faster than 70 miles per hour?
A. 0.83
B. 0.92
C. 0.70
D. 0.67
E. 0.96
(M) 46. Willard has just been given a ten-question multiple choice test in one of his classes.
Each question has five answers only one of which is correct. Since Willard has not
attended class recently, he does not know any of the answers. Assume that Willard
guesses randomly on all ten questions. Find the probability that he will answer, at
most, 3 questions correctly.
A. 0.50
B. 0.80
C. 0.88
D. 0.77
E. 0.70
(M) 47. Assume that 65% of the cars on a freeway are traveling faster than 75 miles per
hour. A random sample of 12 cars was observed under normal driving conditions
with no police car in sight. What is the probability that more than 6 of them were
going faster than 75 miles per hour?
A. 88%
B. 79%
C. 65%
D. 92%
E. 98%
(M) 48. Willard has just been given a 20-question multiple choice test in one of his classes.
Each question has five answers only one of which is correct. Since Willard has not
attended class recently, he does not know any of the answers. Assume that Willard
guesses randomly on all 20 questions. Find the probability that he will answer, at
most, 5 questions correctly.
A. 0.97
B. 0.91
C. 0.80
D. 0.63
E. 0.41
(S) 49. In the past year, Valentino’s Pizza Restaurant grossed more than $1500 a day for
about 70% of its business days. During the next 8 business days, what is the
probability (rounded to two decimal places) that Valentino’s will gross more than
$1500 a day for
(a) 7 or more days?
(b) 5 or more days?
(c) less than 4 days?
(S) 50. In rolling two fair dice simultaneously, the probability of rolling a double (that is,
7
the same number face up on each die), a 7, or an 11 is 18
. If two dice are rolled
simultaneously 5 times, what is the probability (to two decimal places) of this
occurring
(a) 0 times?
(b) 5 times?
(c) at most, 3 times?
(d) more than 3 times?
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Test Item File Understandable Statistics, 7th Edition
(S) 51. The probability that a theater patron will request seating on the main floor is 0.35. A
random sample of 6 patrons call for tickets. Let r be the number who request mainfloor seating.
(a) Find P(r) for r = 0, 1, 2, 3, 4, 5, and 6.
(b) Make a histogram for the r probability distribution.
(c) What is the expected number out of 6 who will request main-floor seating?
(d) Find the standard deviation of r.
(S) 52. Long-term history has shown that 65% of all elected offices in a rural county have
been won by Republican candidates. This year there are 5 offices up for public
election in the county. Let r be the number of public offices won by Republicans.
(a) Find P(r) for r = 0, 1, 2, 3, 4, and 5.
(b) Make a histogram for the r probability distribution.
(c) What is the expected number of Republicans who will win office in the coming
election?
(d) What is the standard deviation of r?
(S) 53. Long-term history has shown that 75% of all elected offices in an urban district have
been won by Democrats. This year there are 8 public offices up for election.
(a) Find the probability that five or more offices are won by Democrats.
(b) Find the probability that all of the offices are won by Democrats.
(c) What is the expected number of Democrats who will win office in this election?
(S) 54. A student has an option of using the American Heritage Dictionary, Webster’s or
Random House in English 103. 1/3 of the students use Webster’s. 2/3 of the
students use one of the other dictionaries.
(a) Find the probability that out of 4 English 103 students, 3 or more use Webster’s.
(b) If 375 students are registered for English 103 next term and each student buys a
dictionary at the bookstore, what is the expected number of Webster’s
dictionaries the bookstore will need?
(S) 55. Jim is an automobile salesman at Courtesy Cars, Incorporated. He has a history of
making a sale for about 15% of all prospective customers that he takes for a test
drive. On most days he takes about 8 prospects for test drives. Let r be the number
of sales on the days when Jim has eight customers.
(a) Find P(r) for r = 0, 1, 2, 3, 4, 5, 6, 7, and 8.
(b) Make a histogram for the r probability distribution.
(c) What is the expected number of cars Jim will sell on the days when he has eight
prospective customers?
(d) What is the standard deviation of the r probability distribution?
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Chapter 5 The Binomial Probability Distribution and Related Topics
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(S) 56. Jim is an automobile salesman at Courtesy Cars, Incorporated. He has a history of
making a sale for about 15% of all prospective customers that he takes for a test
drive. How many prospective customers must Jim have if he wants to be 80% sure
of making at least one sale?
(S) 57. The owners of a motel in Florida have noticed that in the long run about 40% of the
people who stop to inquire about a room for the night actually rent a room.
(a) How many inquiries must the owners answer to be 99% sure of renting at least
one room?
(b) If 25 separate inquiries are made about rooms, what is the expected number of
rentals coming from these inquiries?
(S) 58. The athletic director at Smoky Hills College is recruiting freshman basketball
players. From past experience he knows that the probability that an athlete whom he
contacts will come to Smoky Hills is 75%.
(a) What is the minimum number of players he should contact so that the
probability of at least 6 successful recruits is 90% or higher?
(b) If he contacts eight players, what is the expected number of the eight who will
come to Smoky Hills?
(S) 59. Sharon makes telephone appeals for donations of used clothing and household
goods for a charitable organization. She knows from experience that about 30% of
the calls result in donations.
(a) How many calls must she make to be 90% sure of getting at least two donations.
(b) What is the expected number of donations from 15 calls.
(S) 60. A statewide survey of adults found that only 40% linked the greenhouse effect with
the phenomenon of global warming. Suppose 20 adults of the region were asked
about the greenhouse effect.
(a) What is the probability that fewer than 5 of them linked the greenhouse effect to
global warming?
(b) What is the probability that 16 or more of them linked the greenhouse effect to
global warming?
(c) What is the expected number of people in this group who would see the
connection between the greenhouse effect and global warming?
(S) 61. An insurance company says that 15% of all fires are caused by arson. A random
sample of five fire insurance claims are under study. Let r be the number of claims
in this sample from fires that were started by arson.
(a) Make a histogram for the probability distribution of r.
(b) What is the expected number of arson fires among five fires?
(c) What is the standard deviation of r?
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Test Item File Understandable Statistics, 7th Edition
(S) 62. Safety engineers say that an automobile tire with less than 1/32 of an inch of rubber
tread is not safe. It has been estimated that 10% of all cars in a western state have
unsafe tires. The highway patrol recently stopped 7 cars and did a safety check
including a check of tire tread. Let r be the number of cars in the sample that had
unsafe tires.
(a) Make a histogram for the probability distribution of r.
(b) What is the expected number of cars in the sample with unsafe tires?
(c) What is the standard deviation of r?
(S) 63. Garfield College has an art appreciation course in the humanities program that is
taken on a pass/fail basis. Over a long period of time the art department has
observed that about 80% of the students enrolled in the course pass the course.
Sixteen students are enrolled in a typical section of the course this term.
(a) Find the probability that all of the students in a 16-student section pass the
course.
(b) Find the probability that 10 or fewer of the students in a 16-student section pass
the course.
(c) Find the expected number of students in the section who will pass the course.
(S) 64. A new serum is claimed to be 70% effective in preventing the common cold. A
random sample of 11 people are injected with the serum. The serum is considered
successful if someone who has taken the serum survives the winter without a cold.
(a) Find the probability that the serum is effective for all 11 people.
(b) Find the expected number of people in the sample who survive the winter
without a cold.
(c) Find the standard deviation of this probability distribution.
(S) 65. A biologist has found that 40% of all brown bears are infected with trichinosis.
(a) What is the expected number of infected brown bears in a random sample of 27
brown bears?
(b) What is the standard deviation of the probability distribution?
(S) 66. Salvage Products is a mail-order firm that sells merchandise at a substantial discount
because some of the items are damaged or have missing parts. Customers may not
return the merchandise. The catalogue lists rubber raincoats at 80% off the regular
price. However, many of the raincoats leak. The catalogue states that 40% of the
raincoats in stock leak.
(a) How many of these raincoats should you order to be 97% sure that at least one
does not leak?
(b) If you order 10 raincoats what is the expected number which do not leak?
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Chapter 5 The Binomial Probability Distribution and Related Topics
141
(S) 67. Over the past several years Ms. Carver has determined that she makes a successful
sales call about 20% of the time. What is the minimum number of calls Ms. Carver
must make to be at least 89% sure of making at least one sale?
(S) 68. Derrick scores on 70% of the free throws he attempts in basketball. What is the
minimal number of free throws he must attempt to be at least 89% sure of making at
least 2 of the free throws?
(M) 69. The quality-control inspector of a production plant will reject a batch of automobile
batteries if three or more defectives are found in a random sample of eleven
batteries taken from the batch. Suppose the batch contains 7% defective batteries.
Estimate the mean and standard deviation, µ and ! , of this binomial distribution.
What is the expected number of defective batteries the inspector will find?
A. µ = 0.77, ! = 0.85; the inspector will expect to find 1 defective battery.
B. µ = 0.85, ! = 0.77; the inspector will expect to find 1 defective battery.
C. µ = 0.77, ! = 0.85; the inspector will expect to find 0 defective battery.
D. µ = 0.77, ! = 0.72; the inspector will expect to find 1 defective battery.
E. µ = 0.72, ! = 0.77; the inspector will expect to find 0 defective battery.
(M) 70. The quality-control inspector of a production plant will reject a batch of automobile
batteries if three or more defectives are found in a random sample of eleven
batteries taken from the batch. Suppose the batch contains 7% defective batteries.
What is the probability that the batch will be accepted?
A. 0.97
B. 0.96
C. 0.95
D. 0.99
E. 0.04
(S) 71. The quality-control inspector of a production plant will reject a batch of lamps if
two or more defectives are found in a random sample of six lamps taken from the
batch. Suppose the batch contains 18% defective lamps.
(a) Make a histogram showing the probability of r = 0, 1, 2, 3, 4, 5, 6 defective
lamps in a random sample of six lamps.
(b) Find µ. What is the expected number of defective lamps the inspector will find?
(c) What is the probability (to two decimal places) that the batch will be accepted?
(d) Find ! (to two decimal places).
(S) 72. Laura teaches a pottery class in her studio. She estimates one out of every five
people who call for information about a class will sign up for the class. How many
phone calls must she receive before she is at least 90% sure that at least four people
will sign up for the class?
(S) 73. Ben sells insurance out of an office downtown. He estimates that 30% of the people
he talks to end up buying insurance from him. Based on this estimation, what is the
least number of people Ben should talk to so that his expected number of new
clients number is at least 15?
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(M) 74. Patricia teaches flying lessons at a local airfield. She estimates that 40% of the
people she talks to about flying lessons end up learning to fly with her. What is the
smallest number of people Patricia needs to talk to in order to be at least 90% sure
she will have three or more new flying students?
A. 8
B. 10
C. 12
D. 14
E. 16
(M) 75. A large bank vault has several automatic burglar alarms. The probability is 0.6 that a
single alarm will detect a burglar. How many such alarms should be used to be at
least 90% certain that a burglar trying to enter is detected by at least one alarm?
A. 2
B. 3
C. 4
D. 5
E. 6
(M) 76. A large bank vault has several automatic burglar alarms. The probability is 0.6 that a
single alarm will detect a burglar. If the bank installs eight alarms, what is the
expected number of alarms that would detect a burglar?
A. 2
B. 3
C. 4
D. 5
E. 6
(M) 77. In a binomial distribution with n = 9 trials, the probability of success is p = 0.52.
Find µ and ! to two decimal places.
A. µ = 1.50, ! = 4.32
B. µ = 4.68, ! = 1.50
C. µ = 2.16, ! = 2.25
D. µ = 4.32, ! = 1.50
E. µ = 4.32, ! = 2.25
(S) 78. David is answering the telephone during a radio broadcast fund drive. People phone
in pledges of $5 or more per month. Experience has shown that 45% of the callers
pledge more than $5 per month.
(a) What is the probability that on a given day the first caller to pledge over $5 per
month is the fourth call of the day?
(b) What is the probability that the first caller to pledge over $5 per month will be
among the first three calls?
(S) 79. Richard has volunteered to help with his college’s telefund. His sales experience
leads him to believe that he has a 65% chance of getting the person he calls to
donate to the college. What is the probability that his first donation will not come
before his third call?
(S) 80. Automobiles arrive at the gate of the student parking structure on a Saturday
morning at the rate of 6 per hour.
(a) What is the probability that there will be no cars arriving from 8 A.M. to 9 A.M.
next Saturday morning?
(b) What is the probability that there will be exactly two cars arriving in a 20
minute period on Saturday morning.
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Chapter 5 The Binomial Probability Distribution and Related Topics
143
(S) 81. Inquiries at the reference desk of the library occur at the rate of 4 per hour during
Sunday afternoons.
(a) What is the probability that there will be exactly 5 inquiries in a 2 hour period
on next Sunday afternoon?
(b) What is the probability that there will be no inquiries in a half hour period on a
Sunday afternoon?
(c) What is the probability that there will be at least one inquiry during a half hour
period on Sunday afternoon?
(S) 82. Calls to the homework hotline come in at the rate of 10 per hour on a typical
weekday evening.
(a) What is the probability that there will be fewer than two calls in a 15 minute
period on a weekday evening?
(b) What is the probability that there will be more than two calls in a 15 minute
period on a weekday evening?
(S) 83. A loom which produces plaid wool fabric is known to produce, on the average, one
noticeable flaw per 20 yards of fabric.
(a) What is the probability that there will be exactly two flaws in a twenty-yard
piece of the wool?
(b) What is the probability that there will be no flaws in a five-yard piece of the
wool?
(S) 84. A rare blood condition is found in only 2% of the population.
(a) What is the mean number of people in a random sample of 300 who would have
the blood condition?
(b) Find the probability that no more than 3 people in a sample of 300 people would
have the condition.
(S) 85. Telephone calls come in to the mathematics department office at the rate of 5 per
hour.
(a) Find the probability that there will be no calls in the next ten minutes.
(b) Find the probability that there will be one or more calls in the next 10 minutes.
(S) 86. Donna is a therapist whose clients may page her in case of an emergency. On a
typical weekend she gets on the average 0.2 calls per hour.
(a) Find the probability that Donna gets no calls in a four hour period on a
weekend.
(b) Find the probability that she gets one or more calls in a four hour period on a
weekend.
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Test Item File Understandable Statistics, 7th Edition
(S) 87. Suppose that the average number of customers entering a bank in a 30 minute period
is 8. The bank manager wants to determine the probability that exactly 10 customers
enter the bank in a 30 minute period.
(a) What is the value of λ?
(b) What is the probability that exactly 10 customers enter the bank during a 30
minute period?
(S) 88. The probability that a new medication produces a serious side effect is 0.02.
Estimate the probability that exactly three of 100 patients will experience the side
effect.
(S) 89. The probability of getting a three when you throw a single die is 1/6. You throw the
die repeatedly stopping when you get a three.
(a) What is the probability that you will get a three on or before the third throw?
(b) What is the probability that you will not get a three until after the third throw?
(S) 90. Suppose that the average number of calls received by a business is 4 in a 15-minute
period.
(a) Find the probability that no calls will be received during a 15-minute period.
(b) Find the probability that no calls will be received during a 30-minute period.
(S) 91. On a typical day the probability that one employee will call in sick is 0.04. Estimate
the probability that exactly 5 out of 100 employees will call in sick tomorrow.
(S) 92. Visitors enter an art gallery at the rate of 5 per hour.
(a) What is the mean number of visitors that would come in during a 30 minute
period?
(b) What is the probability that at least one person will come in during a 30 minute
period?
(S) 93. The probability that a new tire will have a blow-out during the first six months of
use is 0.004.
(a) Estimate the probability that exactly 4 tires in a random sample of 800 tires will
have a blow-out during the first six months of use.
(b) Estimate the probability that none of the tires is a random sample of 800 tires
will have a blow-out during the first six months of use.
(S) 94. Lightning strikes occur in the area around the top of Mount Smokey at the average
rate of three per hour during summer afternoons.
(a) Find the probability that there will be at most one lightning strike in this area
during a 2 hour period on a summer afternoon.
(b) Find the probability that there will be exactly 6 lightning strikes in this area
during a 2 hour period on a summer afternoon.
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Chapter 5 The Binomial Probability Distribution and Related Topics
145
(S) 95. Marcella is a real estate agent. On the average, she sells about three houses every
two months.
(a) Find the probability that she will sell exactly one house in the next month.
(b) Find the probability that she will sell two or more houses in the next month.
(S) 96. Snowfalls of 36 inches or more occur in a rural community on the average once
every five years.
(a) Find the probability of having two such storms in a single year.
(b) Find the probability of having more than one such storm in a single year.
(S) 97. The probability of finding a gold nugget in an ore sample in a particular location is
estimated to be 0.0015. 200 independent samples are taken.
(a) Estimate the probability of finding no gold nuggets in the 200 samples.
(b) Estimate the probability of finding one or more gold nuggets in the 200
samples.
(S) 98. The probability of winning the jackpot on a single try on a particular slot machine is
reported to be 0.003.
(a) Estimate the probability of not winning the jackpot on any of 200 independent
tries.
(b) Estimate the probability of winning the jackpot exactly once in 200 independent
tries.
(S) 99. Fred is taking a mathematics course this semester which on average fails about 40%
of students who take it. Let n = 1, 2, 3, … represent the number of times a student
takes this particular math course until the first passing grade is received. Assume the
trials are independent.
(a) Write out a formula for the probability distribution of the random variable n.
(b) What is the probability that Fred passes on the first try (n = 1)?
(c) What is the probability that Fred passes on the second try (n = 2)?
(d) What is the probability that Fred needs three or more tries to pass this particular
mathematics course?
(e) What is the expected number of attempts at this particular mathematics course
Fred must make to pass? Hint: Use µ for the geometric distribution and round.
(S) 100. In order to compare Poisson and binomial distributions, use n = 120, p = 0.05, and
r = 3 to compute (to four decimal places) P(r) using the formula
(a) P(r ) = Cn,r p r (1 ! p ) n!r for the distribution, and
e ! " "r
(b) P(r) =
for Poisson distribution, where ! = np.
r!
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Test Item File Understandable Statistics, 7th Edition
(M)101. In a small town in the midwest United states, 43% of the town’s current residents
were born in the town. Use the geometric distribution to estimate the probability of
not meeting a native to the town until the fourth person you meet.
A. 0.19
B. 0.94
C. 0.43
D. 0.68
E. 0.08
(M)102. In a small town in the midwest United States, 43% of the town’s current residents
were born in the town. Use the geometric distribution to estimate the probability of
meeting a native to the town among the first four people that are met.
A. 0.43
B. 0.68
C. 0.81
D. 0.89
E. 0.94
(M)103. Based on polling results in a particular city, it was found that 7% of cars and trucks
are black in color. Use the Poisson distribution to estimate the probability of
encountering 2 or more black vehicles among 20 that are noticed on the road.
A. 0.59
B. 0.24
C. 0.14
D. 0.76
E. 0.41
(M)104. At the drive-thru window of a local food establishment, it was found that during
slower periods of the day, vehicles visited at the rate of 11 per hour. Use the Poisson
distribution to estimate the probability of three or more vehicles visiting the drivethru within a ten-minute interval during one of these slow periods.
A. 0.24
B. 0.08
C. 0.28
D. 0.33
E. 0.16
(M)105. At a city council mixer, it was commonly known that approximately 65% of the
attendees were members of the Republican Party. Use the geometric distribution to
estimate the probability of not meeting a Republican Party member until you’ve met
three people.
A. 0.03
B. 0.08
C. 0.96
D. 0.23
E. 0.65
(M)106. In 2000, approximately one out of every eight crimes committed in Canada were
violent crimes (Source: www.statcan.ca). Of 50 crimes committed in Canada during
this year, what is the probability, based on the Poisson distribution, that five or more
were violent?
A. 0.85
B. 0.25
C. 0.75
D. 0.15
E. 0.46
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